Question

Suppose you are selling car insurance. You have a customer with the utility function U(w)=600-200 Where w is the customer's wealth. With probability 0.1, this customer is going to get into an accident while driving. If there is an accident, the customer would have to pay for all the damages and would end up with wealth of $1. The customer has a 0.9 probability of not getting into an accident, and if this were to happen, the individual would get to keep his entire wealth of $100. (a) (10 mark) Suppose you want to maximize profit. What price of insurance would you charge the customer for full insurance? That is, what is the customer's maximum willingness to pay for full insurance? (b) (10 mark) What is your expected profit with the insurance price found in (a)? Suppose you are charging $50 for the insurance premium (instead of the price you found in (a). Suppose the probability of accident is q instead of 0.1) (c) (10 mark) What is the smallest value of q so that the customer purchases insurance? Suppose you are charging $50 for the insurance. And suppose the customer who purchased the insurance has a probability of accident, q=1/99, if he drives carefully. In this case (driving carefully), the utility function is the same as before U(w) = 600 - (200/w) If the customer drives recklessly, he gets +1 util because it gives him the thrill U(w)=601-200/w) while increasing 4 to 4 = 1/10. (d) (30 marks) Consider the following three cases. Case C: The customer does not purchases insurance and drives carefully Case A: The customer purchases insurance and drives recklessly Case B: The customer purchases insurance and drives carefully Which of the 3 options will the customer choose? (Hint: compare the EU for each case) To mitigate this moral hazard problem, you introduce deductible of S40 into the insurance policy. If it is mandated by law that all drivers must purchase insurance, (e) (20 marks) With such insurance with a deductible, how will the customer drive? Recklessly or carefully? (Hint: compare the EU for each case)

Answer 1

W U(W) = 600 - 200 probability of accident is o.l probability on not getting, accident is 0.9 Initial wealth is $100. While consumer would end up with $1 ib meet with accident. EU (no insurance) = 0.1XC600 - 200 ) +0.9 (6.06. - .0 - 578. 2 001 Leto be the premium. I too EU (with insurance) = 6oor e TW-P 900 600-100-P 101

100 Goo - 2007, 578.2 » 21.87 20 » 0.10971 100-P 01044 700 p = 4.17 < 100-P p < 99.84 PL 90.83 consumer's maximum willingness to pay is $90083, 2 e) Expected probit = 90.83 - .1 X 99 - $80.93. LE(A)=P-KiLg Tiz porolo out acci dent L= amount out lochy C) Let, insurance premium is $50. probability of accident is a doua .: we have a +[600-400) + V (600- 2,00) (600-90005), • 18(800 - 200 ) atw (600-400 5967 40 t 5980 7 596-407 5980 5567, 5989 = 0.927 U a< 0.92 . The smallest value a iy 0.92 so that consumer purchase insurance. 1919